Correlated Markov Quantum Walks

TL;DR

This paper analyzes the long-term behavior of correlated quantum walks on ^d, showing that the averaged position distribution exhibits drift and diffusion, with explicit formulas for the diffusion matrix and large deviations, extending understanding of quantum dynamics with Markovian randomness.

Contribution

It introduces a spectral analysis framework for correlated quantum walks with Markovian coin updates, deriving explicit formulas for drift, diffusion, and large deviations in the averaged distribution.

Findings

The averaged distribution exhibits a linear drift proportional to time.

Centered distribution shows diffusive behavior with a computable diffusion matrix.

Large deviations principles are established for the averaged distribution.

Abstract

We consider the discrete time unitary dynamics given by a quantum walk on $\Z^d$ performed by a particle with internal degree of freedom, called coin state, according to the following iterated rule: a unitary update of the coin state takes place, followed by a shift on the lattice, conditioned on the coin state of the particle. We study the large time behavior of the quantum mechanical probability distribution of the position observable in $\Z^d$ for random updates of the coin states of the following form. The random sequences of unitary updates are given by a site dependent function of a Markov chain in time, with the following properties: on each site, they share the same stationnary Markovian distribution and, for each fixed time, they form a deterministic periodic pattern on the lattice. We prove a Feynman-Kac formula to express the characteristic function of the averaged distribution over the randomness at time $n$ in terms of the nth power of an operator $M$. By analyzing the spectrum of $M$, we show that this distribution posesses a drift proportional to the time and its centered counterpart displays a diffusive behavior with a diffusion matrix we compute. Moderate and large deviations principles are also proven to hold for the averaged distribution and the limit of the suitably rescaled corresponding characteristic function is shown to satisfy a diffusion equation. An example of random updates for which the analysis of the distribution can be performed without averaging is worked out. The random distribution displays a deterministic drift proportional to time and its centered counterpart gives rise to a random diffusion matrix whose law we compute. We complete the picture by presenting an uncorrelated example.